Enumerative questions on rooted weighted necklaces (Q1271869)

A rooted necklace is a graph consisting of \(k\) vertices \(v_0,v_1, \dots,v_{k-1}\), of which \(v_0\) is distinguished as the root, joined cyclically: the edges are \(\{v_{i-1},v_i\}\), \(1\leq i\leq k-1\), and \(\{v_{k-1},v_0\}\). A rooted necklace is weighted by attributing an integer (a number of chips) to each vertex. In [\textit{B. Cipra}, Proposed problem 1388, Math. Mag. 65, No. 1, 56 (1992)] the following problem was posed. Suppose you start with a \(k\)-vertex necklace rooted at \(v_0\) and \(n\) chips on each vertex. Take all the chips off \(v_0\) and distribute them cyclically: one chip at a time to \(v_1,v_2,\dots\) until you run out of chips. Relabel the vertices so that the last vertex \(v_r\) to receive a chip becomes the new root and is labeled \(v_0\), \(v_{r+1}\) is labeled \(v_1\) and so on. Then repeat the process until a rooted weighted necklace gets created that has been created before. How many different rooted weighted necklaces will get created? Cipra and several others solved this problem for \(k=2\) and general \(n\). The paper under review solves it for \(n=1\) and general \(k\). The solution for general \(n\) and \(k\) is still an open problem.